Problem: Find the number of ordered 17-tuples $(a_1, a_2, a_3, \dots, a_{17})$ of integers, such that the square of any number in the 17-tuple is equal to the sum of the other 16 numbers.
Explanation: Let $S = a_1 + a_2 + a_3 + \dots + a_{17}.$  Then from the given condition,
\[a_i^2 = S - a_i\]for all $1 \le i \le 17.$  In other words, each $a_i$ is a root of
\[x^2 + x - S = 0.\]This quadratic has at most two roots, which means that there are at most two different values among the $a_i,$ for any particular 17-tuple.

Suppose that all the $a_i$ are equal, say
\[a = a_1 = a_2 = a_3 = \dots = a_{17}.\]Then $S = 17a,$ so from the equation $x^2 + x - S = 0,$
\[a^2 + a - 17a = 0.\]Then $a^2 - 16a = a(a - 16) = 0,$ so $a = 0$ or $a = 16.$

Otherwise, there are exactly two different values among the $a_i,$ say $a$ and $b.$  Suppose that $n$ of the $a_i$ are equal to $a,$ so the remaining $17 - n$ values are equal to $b,$ where $1 \le n \le 16.$  Then
\[S = na + (17 - n) b.\]Since $a$ and $b$ are the roots of $x^2 + x - S = 0,$ by Vieta's formulas, $a + b = -1$ and $ab = -S.$  Hence,
\[na + (17 - n) b = -ab.\]From $a + b = -1,$ $b = -a - 1.$  Substituting, we get
\[na + (17 - n)(-a - 1) = -a(-a - 1).\]This simplifies to
\[a^2 + (-2n + 18) a - n + 17 = 0. \quad (*)\]Since $a$ is an integer, the discriminant of this polynomial must be a perfect square.  Thus,
\[(-2n + 18)^2 - 4(-n + 17) = 4n^2 - 68n + 256 = 4(n^2 - 17n + 64)\]is a perfect square, which means $n^2 - 17n + 64$ is a perfect square.

Checking all values in $1 \le a \le 16,$ we find that $n^2 - 17n + 64$ is a perfect square only for $n = 5$ and $n = 12.$

For $n = 5,$ equation $(*)$ becomes
\[a^2 + 8a + 12 = (a + 2)(a + 6) = 0,\]so $a = -2$ or $a = -6.$  The respective values of $b$ are $b = 1$ and $b = 5.$

So one possibility is that five of the $a_i$ are equal to $-2,$ and the remaining 12 are equal to 1.  There are $\binom{17}{5} = 6188$ 17-tuples of this form.  Another possibility is that five of the $a_i$ are equal to $-6,$ and the remaining 12 are equal to 5.  There are $\binom{17}{5} = 6188$ 17-tuples of this form.

The case $n = 12$ leads to the same possibilities.  Therefore, the total number of 17-tuples is $2 + 6188 + 6188 = \boxed{12378}.$